Extension to the Hybrid Symbolic-Numeric Method for Investigating Identifiability Diana Cole, SE@K, University of Kent, UK Rémi Choquet, CEFE, CNRS, France

Occupancy Model – Obvious Example of Parameter Redundancy Each site visited just once. 𝜓 probability site is occupied, 𝑝 probability species is detected. Either species observed at site with probability: 𝜓𝑝 Or species not observed at site with probability: 𝜓 1−𝑝 +1−𝜓 =1−𝜓𝑝 Can only estimate 𝛽=𝜓𝑝 rather than 𝜓 and 𝑝. Model is parameter redundant or parameters are non-identifiable. (Solution: MacKenzie et al, 2002, robust design; visit sites several times in one season). Species present but not seen Species not present

Detecting Parameter Redundancy Method Accurate Identifiable Parameters Estimable Parameter Combinations General Results Complex Models Automatic Numeric (eg Viallefont et al, 1998) ×  Symbolic (Catchpole and Morgan, 1997) Extended Symbolic (Cole et al, 2010) Hybrid Symbolic-Numeric (Choquet and Cole, 2012) Extended Hybrid ?

Occupancy Model - Likelihood Surface

Occupancy Model - Likelihood Profile Flat profile = Parameter Redundant

Occupancy Model - Likelihood Profile Line is 𝑝= 𝛽 𝜓 = 0.5 𝜓 ⇒0.5=𝜓𝑝 Profile gives the estimable parameter combinations

Extended Hybrid Method – Subset Profiling Eisenberg and Hayashi (2014) use subset profiling to identify estimable parameter combinations manually in compartment modelling. Combine hybrid method (Choquet and Cole, 2012) with subset profiling (Eisenberg and Hayashi, 2014) and extend to automatically detect relationships between confounded parameters. Step 1: Check whether model is parameter redundant using hybrid method and ignore any parameters that are individually identifiable. Step 2: Find subsets of parameters that are confounded, using subset profiling. Step 3: For each subset produce a likelihood profile for each pair of parameters. Fix one parameter, 𝜃 1,𝑖 , and store the MLE of the other parameter, 𝜃 2,𝑖 (𝑖=1,…,𝑛).

Extended Hybrid Method – Subset Profiling Step 4: Identify the relationships. Find 𝛽 1 , 𝛽 2 , 𝛽 3 , 𝛽 4 that minimise 𝑖=1 𝑛 𝜃 2,𝑖 − 𝛽 1 + 𝛽 2 𝜃 1,𝑖 𝛽 3 + 𝛽 4 𝜃 1,𝑖 2 to identify most common relationships. (Should equal 0). Occupancy Example: 𝜃 1 =𝜓, 𝜃 2 =𝑝, 𝛽 1 =0.5, 𝛽 2 =0, 𝛽 3 =0, 𝛽 4 =1 𝑝− 0.5+0×𝜓 0+1×𝜓 =𝑝− 0.5 𝜓 =0⇒𝑝𝜓=0.5 Step 5: Turn relationships between pairs of parameters into estimable parameter combinations. Steps 1-4 can be programmed to be automatic, step 5 not yet.

Mark-Recovery Example Animals marked and recovered dead. Probability animals marked in year 𝑖 and recovered in year 𝑗: 𝑃= 1− 𝜙 1,1 𝜆 1 𝜙 1,1 1− 𝜙 2 𝜆 2 𝜙 1,1 𝜙 2 1− 𝜙 3 𝜆 3 1− 𝜙 1,2 𝜆 1 𝜙 1,2 1− 𝜙 2 𝜆 2 1− 𝜙 1,3 𝜆 1 𝜙 1,𝑡 probability animal survives 1st year at time t. 𝜙 𝑖 probability an animal age 𝑖 survives their 𝑖th year. 𝜆 𝑖 reporting probability for animal aged 𝑖. Step 1: Hybrid method can be use to show model is parameter redundant. 𝜙 1,1 , 𝜙 1,2 , 𝜙 1,3 , 𝜆 1 are individually identifiable. Non-identifiable parameters are 𝜙 2 , 𝜙 3 , 𝜆 2 , 𝜆 3 . Step 2: Subsets are: { 𝜙 3 , 𝜆 3 }, { 𝜙 2 , 𝜙 3 , 𝜆 2 }, { 𝜙 2 , 𝜆 2 , 𝜆 3 }

Mark-Recovery Example Step 3: x profiles for subset { 𝜙 3 , 𝜆 3 }: Step 4: fitted relationship: 𝜆 3 − 0.0736−0.0000 𝜙 3 1.2267−1.2268 𝜙 3 =0⇒ 1− 𝜙 3 𝜆 3 =0.06

Mark-Recovery Example Step 3: x profiles for subset { 𝜙 2 , 𝜙 3 , 𝜆 2 } Step 4: fitted relationships: 𝜙 3 − −0.1800+0.9999 𝜙 2 0.0000+0.9999 𝜙 2 =0⇒ 𝜙 2 1− 𝜙 3 =0.18 𝜆 2 − 0.1200+0.0000 𝜙 2 0.9999−0.9999 𝜙 2 =0⇒ 1− 𝜙 2 𝜆 2 =0.12 𝜆 2 − −5.9997+5.9998 𝜙 3 −40.9973+49.9970 𝜙 3 =0⇒ 𝜆 2 −41+50 𝜙 3 −6 𝜙 3 =−6 (complex)

Mark-Recovery Example Step 3: x profiles for subset { 𝜙 2 , 𝜆 2 , 𝜆 3 } Step 4: fitted relationships: 𝜆 2 − 0.1200+0.0000 𝜙 2 0.9999−0.9999 𝜙 2 =0⇒ 1− 𝜙 2 𝜆 2 =0.12 𝜆 3 − 0.1472−0.0000 𝜙 2 0.0000+1.2267 𝜙 2 =0⇒ 𝜙 2 𝜆 3 =0.12 𝜆 3 − −0.0001+2.9998 𝜆 2 −2.9999+24.9979 𝜆 2 =0⇒ 𝜆 2 𝜆 3 𝜆 2 + 𝜆 3 = 3 25 (complex)

Mark-Recovery Example Step 5: There could be more than one way of parameterising the estimable parameter combinations. Look for the simplest. From Step 1, we can deduce that we need to find 2 combinations of parameters. From Step 2, we can also deduce that 𝜙 2 , 𝜙 3 , 𝜆 3 are related. Relationships that involve 𝜙 2 , 𝜙 3 , 𝜆 3 : 𝜙 2 1− 𝜙 3 =0.18, 𝜙 2 𝜆 3 =0.12, 1− 𝜙 3 𝜆 3 =0.06 are consistent with: 𝝓 𝟐 𝟏− 𝝓 𝟑 𝝀 𝟑 1− 𝜙 2 𝜆 2 =0.12 is consistent with: 𝟏− 𝝓 𝟐 𝝀 𝟐 Estimable parameter combinations: 𝝓 𝟐 𝟏− 𝝓 𝟑 𝝀 𝟑 & 𝟏− 𝝓 𝟐 𝝀 𝟐 (Agrees with symbolic results - see Cole et al, 2012.) (The other complex relationships can be shown, with a bit of algebra to be consistent with estimable parameter combinations.)

General Results In the symbolic method general results can be found using the exteion theorem (Catchpole and Morgan, 1997, Cole et al, 2010). This theorem can be extended to the Hybrid method. Mark Recovery Example: Reparameterise in terms of estimable parameters, 𝜙 1,1 , 𝜙 1,2 , 𝜙 1,3 , 𝜆 1 , 𝑏 1 = 1− 𝜙 2 𝜆 2 and 𝑏 2 = 𝜙 2 1− 𝜙 3 𝜆 3 . Hybrid method shows that we can estimate 𝜙 1,1 , 𝜙 1,2 , 𝜙 1,3 , 𝜆 1 , 𝛽 1 , 𝛽 2 . Adding an extra year of ringing and recovery adds extra parameters 𝜙 1,4 and 𝑏 3 = 𝜙 2 𝜙 3 1− 𝜙 4 𝜆 4 . Hybrid method shows that we can also estimate 𝜙 1,4 , 𝑏 3 . By the extension theorem, for n years of ringing and recovery, the model is always parameter redundant. Estimable parameter combinations are 𝜙 1,1 , 𝜙 1,2 ,…, 𝜙 1,𝑛 , 𝜆 1 , 1− 𝜙 2 𝜆 2 , 𝜙 2 1− 𝜙 3 𝜆 3 ,…, 𝑖=2 𝑛−1 𝜙 𝑖 1− 𝜙 𝑛 𝜆 𝑛

Identifiable Parameters Estimable Parameter Combinations Conclusion Method Accurate Identifiable Parameters Estimable Parameter Combinations General Results Complex Models Automatic Numeric (eg Viallefont et al, 1998) ×  Symbolic (Catchpole and Morgan, 1997) Extended Symbolic (Cole et al, 2010) Hybrid Symbolic-Numeric (Choquet and Cole, 2012) Extended Hybrid Semi

References Catchpole, E. A. and Morgan, B. J. T. (1997) Detecting parameter redundancy Biometrika, 84, 187-196 Choquet, R. and Cole, D.J. (2012) A Hybrid Symbolic-Numerical Method for Determining Model Structure. Mathematical Biosciences, 236, p117. Cole, D. J. and Morgan, B. J. T. (2010) A note on determining parameter redundancy in age-dependent tag return models for estimating fishing mortality, natural mortality and selectivity, JABES, 15, 431-434. Cole, D. J., Morgan, B. J. T., Catchpole, E. A. and Hubbard, B.A. (2012) Parameter redundancy in mark-recovery models, Biometrical Journal, 54, 507-523. Eisenberg, M. C. and Hayashi, M. A. L (2014) Determining identifiable parameter combinations using subset profiling, Mathematical Biosciences, 256, 116–126. MacKenzie, D. I., et al (2002) Estimating site occupancy rates when detection probabilities are less than one. Ecology, 83, 2248-2255. Viallefont, A., et al (1998) Parameter identifiability and model selection in capture-recapture models: A numerical approach. Biometrical Journal, 40, 313-325.